Main Effects and Interaction Effects Factorial Design Main Effects and Interaction Effects

Pre-test Determine the appropriateness of manipulations (operationalizations of independent variables) Test the validity and reliability of the measurement scales

Pilot Study Use a small group of participants to test whether the questionnaire is clear and understandable Use the pilot study data to provide some intuitive observations about whether the experiment could work

Factorial Design Much experimental psychology asks the question: What effect does a single independent variable have on a single dependent variable? It is quite reasonable to ask the following question as well. What effects do multiple independent variables have on a single dependent variable? Designs which include multiple independent variables are known as factorial designs.

Factorial Design All combinations of two or more values of two or more IVs are created Can be tested within, between or mixed Reasons for factorial designs Efficiency – additional Ivs are included Helps rule out several hypotheses at the same time Allows the researcher to discover interactions amongst the variables Allows the researcher to include one or more nuisance variables in the design

Picture of Toy Factorial Design Bright colour Dull Colour Not Noisy Noisy Soft Not Soft

Describing Toy Factorial Design All possible combinations of soft, noisy and colourful toys in a 2 x 2 x 2 design – i.e., three IVs 2 x 2 x 2 = 8 cells Another factor (IV) with two levels One factor (IV) with 2 levels

An Example Factorial Design If we were looking at GENDER and TIME OF EXAM, these would be two independent factors GENDER would only have two levels: male or female TIME OF EXAM might have multiple levels, e.g. morning, noon or night This is a factorial design

Examples If there are 2 levels of the first IV and 3 levels of the second IV It is a 2x3 design E.G.: coffee drinking x time of day Factor coffee has two levels: cup of coffee or cup of water Factor time of day has three levels: morning, noon and night If there are 3 levels of the first IV, 2 levels of the second IV and 4 levels of the third IV It is a 3x2x4 design E.G.: coffee drinking x time of day x exam duration Factor coffee has three levels: 1 cup, 2 cup 3 cups Factor time of day has two levels: morning or night Factor exam duration has 4 levels: 30min, 60min, 90min, 120min

Difficulties with Factorial Designs More complex, more time Difficult to analyze if some data is missing Need more Ss for each factor, for each extra level Interactions, especially higher order interactions (3, 4 or higher), can be difficult to understand

Main Effects and Interaction Effects The effect of a single variable is known as a main effect The effect of two variables considered together is known as an interaction For the two-way between groups design, an F-ratio is calculated for each of the following: The main effect of the first variable The main effect of the second variable The interaction between the first and second variables

Analysis of a 2-way Between-Subjects Design Using ANOVA To analyse the two-way between groups design we have to follow the same steps as the one-way between groups design State the Null Hypotheses Partition the Variability Calculate the Mean Squares Calculate the F-Ratios

Null Hypotheses There are 3 null hypotheses for the two-way (between groups design. The means of the different levels of the first IV will be the same, e.g. The means of the different levels of the second IV will be the same, e.g. The differences between the means of the different levels of the interaction are not the same, e.g.

An Example Null Hypothesis for an Interaction The differences betweens the levels of factor A are not the same.

Partitioning the Variability If we consider the different levels of a one-way ANOVA then we can look at the deviations due to the between groups variability and the within groups variability. If we substitute AB into the above equation we get This provides the deviations associated with between and within groups variability for the two-way between groups design.

Partitioning the Variability (Cont.) The between groups deviation can be thought of as a deviation that is comprised of three effects. In other words the between groups variability is due to the effect of the first independent variable A, the effect of the second variable B, and the interaction between the two variables AxB.

Partitioning the Variability The effect of A is given by Similarly the effect of B is given by The effect of the interaction AxB equals which is known as a residual

The Sum of Squares The sums of squares associated with the two-way between groups design follows the same form as the one-way We need to calculate a sum of squares associated with the main effect of A, a sum of squares associated with the main effect of B, a sum of squares associated with the effect of the interaction. From these we can estimate the variability due to the two variables and the interaction and an independent estimate of the variability due to the error.

The Mean Squares In order to calculate F-Ratios we must calculate an Mean Square associated with The Main Effect of the first IV The Main Effect of the second IV The Interaction. The Error Term

The mean squares The main effect mean squares are given by: The interaction mean squares is given by: The error mean square is given by:

The F-ratios The F-ratio for the first main effect is: The F-ratio for the second main effect is: The F-ratio for the interaction is:

An Example 2x2 Between-Groups ANOVA Factor A - Lectures (2 levels: yes, no) Factor B - Worksheets (2 levels: yes, no) Dependent Variable - Exam performance (0…30) Mean Std Error LECTURES WORKSHEETS yes 19.200 2.04 no 25.000 1.23 16.000 1.70 9.600 0.81

Results of ANOVA When an analysis of variance is conducted on the data, the following results are obtained Source Sum of Squares df Mean Squares F p A (Lectures) 432.450 1 37.604 0.000 B (Worksheets) 0.450 0.039 0.846 AB 186.050 16.178 0.001 Error 184.000 16 11.500

What Does It Mean? - Main effects A significant main effect of Factor A (lectures) “There was a significant main effect of lectures (F1,16=37.604, MSe=11.500, p<0.001). The students who attended lectures on average scored higher (mean=22.100) than those who did not (mean=12.800). No significant main effect of Factor B (worksheets) “The main effect of worksheets was not significant (F1,16=0.039, MSe=11.500, p=0.846)”

What Does It Mean? - Interaction A significant interaction effect “There was a significant interaction between the lecture and worksheet factors (F1,16=16.178, MSe=11.500, p=0.001)” However, we cannot at this point say anything specific about the differences between the means unless we look at the null hypothesis Many researches prefer to continue to make more specific observations. Mean Std Error LECTURES WORKSHEETS yes 19.200 2.04 no 25.000 1.23 16.000 1.70 9.600 0.81

Analytic Comparisons in General If there are more than two levels of a Factor And, if there is a significant effect (either main effect or simple main effect) Analytical comparisons are required. Post hoc comparisons include Tukey tests, Scheffé test or t-tests (Bonferroni corrected).

Assignment to conditions 2 X 2 3 X 3 X 2 Assignment to conditions Randomized group—completely randomized factorial Matched factorial design—form groups based on matching and randomly assign to conditions

The Purpose of Factorial Models Two or more variables at once Combinations of variables Main effects The effect of one independent variable, averaged over all levels of another independent variable The effect of one independent variable, ignoring the other independent variables

Interaction Main effects effect of one independent variable depends upon the levels of another independent variable the effects of one variable varies over the levels of another independent variable Main effects Model Stimulus

Actual interaction found Three hypotheses Main effects Stimulus Model emotion Interaction—Stimulus X model emotion Actual interaction found Results from similar study

Interactions Graphic representation Main effects conditioned by interaction

Mood and Memory

Factorial Design Analysis 2 X 2 Three questions Two main effects Interaction Source table Source SS df MS F Significance

Possible Patterns No effects

Main Effect No Interaction

Two Main Effects No Interaction

Interaction Only

Main Effect and Interaction

Two Main effects and Interaction

Interactions Interactions Antagonistic Synergistic Ceiling effect

Higher Order Designs Extensions within a two way design More levels 2 X 3 3 X 3 Higher order designs More factors 2 X 2 X 2 Limitations Interpretation of higher order interactions Number of conditions and participants 3 X 3 X 3 X 3 = 81 cells

Higher order interactions

What Is Multivariate Analysis of Variance (MANOVA)? Univariate Procedures For Assessing Group Differences The t Test Analysis of Variance (ANOVA) Multivariate Analysis of Variance (MANOVA) The Two-Group Case: Hotelling's T2 The k-Group Case: MANOVA

Group Comparison Analyses ANOVA ANCOVA MANOVA MANCOVA